The scattering parameters and directional coupler analysis of characteristically terminated asymmetric coupled transmission lines in an inhomogeneous medium

The scattering matrix of asymmetric coupled two-line structures in an inhomogeneous medium terminated in a set of impedances that are equal to the characteristic impedances of the individual, uncoupled lines is derived in terms of the coupled-mode parameters. It is shown that the structures can compose an ideal backward-coupling directional coupler, perfectly matched and isolated at all frequencies, if the inductive k_L and capacitive k_C coupling coefficients of the coupled lines are equal. The effect of the nonideal equalization of the coupling coefficients on the coupler critical parameters is investigated. The normal-mode parameters (mode numbers and mode impedances) in the proximity of the point when k_L=k_C and at that point are also examined. Numerical results confirm the validity of the analysis and prove the possibility of an asymmetrical coupler design with very high directivity